Introduction

SLOPE (Sorted L-One Penalized Estimation) is a method for estimating the parameter \(\beta\) in a parametric statistical model.

  • Similar to LASSO, this algorithm incurs a penalty term based on the \(\ell_{1}\) norm of the estimator \(\hat{\beta}\).
  • Unlike LASSO, SLOPE does not use a constant term \(\lambda\) to calculate the penalty which is applied to the model fit.

Comparison of \(\ell_{1}\) Penalties

  • The penalty term in LASSO regression is \(\lambda\sum_{i=1}^{p}\left\vert\hat{\beta}_{i}\right\vert\).
  • The SLOPE penalty is given by \(\sum_{i=1}^{p}\lambda_{i}\left\vert\hat{\beta}_{(i)}\right\vert\).
    • In this equation, \(\lambda_{1} \ge \lambda_{2} \ge \dots \ge \lambda_{p} \ge 0\), and the elements of \(\hat{\beta}\) are sorted so that \(\left\vert\hat{\beta}_{(1)}\right\vert \ge \dots \ge \left\vert\hat{\beta}_{(p)}\right\vert\).

Paper Ideas

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Experiments

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Proposal

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